# How can I calculate the rate of decay of a radioactive element?

Mar 12, 2015

The decay of a radioactive element is a random process which is governed by the laws of chance.

The rate of decay only depends on the number of undecayed atoms.

This means that the more atoms of a radioactive element you have in your sample, the more chance a decay event will occur in that sample.

This is a 1st order process (which you may have met if you have studied chemical kinetics) for which:

$- R a t e \propto N$

The minus sign shows N is decreasing with time

We can write this as:

$- R a t e = \lambda N$

Where $\lambda$ is the decay constant, which is a constant for a particular isotope.

A process like this follows exponential decay which means that the time taken for half the original sample to decay is a constant.

This is termed the half - life or ${t}_{\frac{1}{2}}$.

Here is an example for the decay of carbon - 14:

It can be shown that ${t}_{\frac{1}{2}} = \frac{0.693}{\lambda}$

Let's see how we can use this to calculate the rate of decay from this example:

"What is the rate of decay of 1.00g of radon - 224 which has a half - life of 55s?"

Rearranging we get:

lambda=(0.693)/(t_(1/2)

$\lambda = \frac{0.693}{55} = 0.0126 {s}^{- 1}$

$R = - l a m \mathrm{da} N$

$R = - 0.0126 \times 1.00 = - 0.0126 \text{g/s}$

(I have used grams and not atoms as these are proportional and is reflected in the answer's units.)